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# Statistical Inference Statistical inference is the process of drawing conclusions about a population based on a sample of data. ## Point Estimation A point estimate is a single value that is used to estimate a population parameter. ### Method of Moments The method of moments is a technique fo...

# Statistical Inference Statistical inference is the process of drawing conclusions about a population based on a sample of data. ## Point Estimation A point estimate is a single value that is used to estimate a population parameter. ### Method of Moments The method of moments is a technique for estimating population parameters by equating sample moments with population moments and solving for the parameters. Let $X_1, X_2,..., X_n$ be a random sample from a population with probability density function $f(x; \theta_1, \theta_2,..., \theta_k)$, where $\theta_1, \theta_2,..., \theta_k$ are the parameters to be estimated. The $j$th population moment is defined as $E(X^j)$, and the $j$th sample moment is defined as $\frac{1}{n}\sum_{i=1}^{n}X_i^j$. To estimate the parameters $\theta_1, \theta_2,..., \theta_k$, we equate the first $k$ population moments to the corresponding sample moments and solve for the parameters. ### Maximum Likelihood Estimation The maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model. When applied to a data set, MLE chooses the set of parameters that maximizes the likelihood function. Given a random sample $X_1, X_2,..., X_n$ from a population with probability density function $f(x; \theta)$, the likelihood function is defined as: $L(\theta; x_1, x_2,..., x_n) = \prod_{i=1}^{n}f(x_i; \theta)$ The MLE of $\theta$ is the value that maximizes the likelihood function. **Example:** Suppose we have a random sample $X_1, X_2,..., X_n$ from a normal distribution with mean $\mu$ and variance $\sigma^2$. The likelihood function is: $L(\mu, \sigma^2; x_1, x_2,..., x_n) = \prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x_i - \mu)^2}{2\sigma^2}}$ To find the MLE of $\mu$ and $\sigma^2$, we take the logarithm of the likelihood function and maximize it with respect to $\mu$ and $\sigma^2$.